Proof of a Folklore Julia Set Connectedness Theorem and Connections with Elliptic Functions
نویسنده
چکیده
We prove the following theorem about Julia sets of the maps fn,p,γ(z) = z n + γ zp , for integers n, p ≥ 2, γ ∈ C by using techniques developed for the Weierstrass elliptic ℘ function and adapted to this setting. Folklore connectedness theorem: If fn,p,γ has a bounded critical orbit, then J(fn,p,γ) is connected. This is related to connectivity results by the author and others about J(℘), where ℘ denotes the Weierstrass elliptic ℘ function, especially where the period lattice has some symmetry. We illustrate several further connections between the dynamics of some specific elliptic functions and the family fn,p,γ for some values of n and p.
منابع مشابه
Some local fixed point results under $C$-class functions with applications to coupled elliptic systems
The main objective of the paper is to state newly fixed point theorems for set-valued mappings in the framework of 0-complete partial metric spaces which speak about a location of a fixed point with respect to an initial value of the set-valued mapping by using some $C$-class functions. The results proved herein generalize, modify and unify some recent results of the existing literature. As an ...
متن کاملExtension of the Douady-Hubbard's Theorem on Connectedness of the Mandelbrot Set to Symmetric Polynimials
متن کامل
Diagonal arguments and fixed points
A universal schema for diagonalization was popularized by N.S. Yanofsky (2003), based on a pioneering work of F.W. Lawvere (1969), in which the existence of a (diagonolized-out and contradictory) object implies the existence of a fixed-point for a certain function. It was shown that many self-referential paradoxes and diagonally proved theorems can fit in that schema. Here, we fi...
متن کاملP2-CONNECTEDNESS IN L-TOPOLOGICAL SPACES
In this paper, a certain new connectedness of L-fuzzy subsets inL-topological spaces is introduced and studied by means of preclosed sets. Itpreserves some fundamental properties of connected set in general topology.Especially the famous K. Fan’s Theorem holds.
متن کاملMATRIX VALUATION PSEUDO RING (MVPR) AND AN EXTENSION THEOREM OF MATRIX VALUATION
Let R be a ring and V be a matrix valuation on R. It is shown that, there exists a correspondence between matrix valuations on R and some special subsets ?(MVPR) of the set of all square matrices over R, analogous to the correspondence between invariant valuation rings and abelian valuation functions on a division ring. Furthermore, based on Malcolmson’s localization, an alternative proof for t...
متن کامل